Let $M \subseteq \mathbb{R}$ be a compact metric space and let $(u_n)$ and $(v_n)$ be two sequences such that $\lim(u_n - v_n) = 0$.
Show that there exist subsequences of $(u_n)$ and $(v_n)$ such that they have the same limit ?
I think we would need the Bolzano-Weierstrass theorem but I clearly don't know where to start.
hint
As a sequence in a compact, $(u_n)$ has a convergent subsequence $ (u_{\phi(n)}) $ with $L$ as its limit. but
$$\lim_{n\to+\infty}(v_{\phi(n)}-u_{\phi(n)})=0$$
and
$$v_{\phi(n)}=(v_{\phi(n)}-u_{\phi(n)})+u_{\phi(n)}$$
thus
$$\lim_{n\to+\infty}v_{\phi(n)}=0+L=L$$